https://www.youtube.com/watch?v=U1HbB0ATZ_A
In Bayesian probability theory, if the posterior distribution p(θ | x) is in the same probability distribution family as the prior probability distribution p(θ), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function p(x | θ).
https://www.youtube.com/watch?v=7mZksQ24MlI
In this case, a Normal prior, turns to a Normal posterior, and a prior Beta turns to a posterior Beta.
https://www.youtube.com/watch?v=vJ-NqIgYJyY
Reference: (Wolpert, n.d.)
wait
https://bookdown.org/kevin_davisross/bayesian-reasoning-and-methods/poisson.html
A discrete random variable Y has a Poisson distribution with parameter \(\theta>0\) if its probability mass function satisfies \[ \begin{align*} f(y|\theta) & = \frac{e^{-\theta}\theta^y}{y!}, \quad y=0,1,2,\ldots \end{align*} \] If \(Y\) has a \(Poisson(\theta)\) distribution then
\[ \begin{align*} E(Y) & = \theta\\ Var(Y) & = \theta \end{align*} \]
Question from Exercice 6 of classes.
If we look at the distribution of gamma, we can conclude it’s a gamma distribution with parameters: \[\theta \sim \Gamma (\mu h, h)\]
Question from Exercice 6 of classes.
See:
* Poisson-Gamma conjugate
If \(\mu = h = 2\), \(\sum_{i=1}^{n} x_i = 18\) and \(n=6\) then the \(a posteriori\) distribution of \(\theta\) is
mu_prior = 2
h_prior = 2
alpha_prior = mu_prior*h_prior
beta_prior = h_prior
sum_xi_1 = 18
n_1 = 6
alpha_posterior = alpha_prior + sum_xi_1
beta_posterior = beta_prior + n_1
cat("Alpha Posterior:", alpha_posterior, "\nBeta Posterior", beta_posterior)
## Alpha Posterior: 22
## Beta Posterior 8
The a posteriori distribution is a \(Gamma(22,8)\)
See:
* Poisson-Gamma conjugate
The a priori distribution is a \(Gamma(2, 1)\) for \(\theta_1\) and \(\theta_2\).
Question from Exercice 6 of classes.
e_mean = 22 / 8
e_var = 22 / ((8)**2) ## 22/64
cat("Expected Mean:", e_mean, "\nExpected Variance", e_var)
## Expected Mean: 2.75
## Expected Variance 0.34375
See:
* Credibility Intervals
So here, to calculate the credibility interval, we do the following
alpha = 22; beta = 8 # parameters of the posterior distribution (Gamma)
qinf_gamma = qgamma(p=0.025,shape=alpha,rate=beta,lower.tail=T)
qsup_gamma = qgamma(p=0.025,shape=alpha,rate=beta,lower.tail=F)
cat("Interval: [",qsup_gamma,",",qinf_gamma,"]")
## Interval: [ 4.012591 , 1.72341 ]
## Alternatively with Gamma
qsup2_gamma = qgamma(p=0.975,shape=alpha,rate=beta,lower.tail=T)
cat("Interval: [",qsup2_gamma,",",qinf_gamma,"]")
## Interval: [ 4.012591 , 1.72341 ]
Visually this means:
TODO: Interpretation of this interval